In this paper we consider a model shape optimization problem. The state variable solves an elliptic equation on a domain with one part of the boundary described as the graph of a control function. We prove higher regularity of the control and develop a priori error analysis for the finite element discretization of the shape optimization problem under consideration. The derived a priori error estimates are illustrated on two numerical examples.
Mots-clés : shape optimization, existence and convergence of approximate solutions, error estimates, finite elements
@article{M2AN_2013__47_6_1733_0,
author = {Kiniger, Bernhard and Vexler, Boris},
title = {\protect\emph{A priori }error estimates for finite element discretizations of a shape optimization problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1733--1763},
publisher = {EDP-Sciences},
volume = {47},
number = {6},
year = {2013},
doi = {10.1051/m2an/2013086},
zbl = {1283.49051},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013086/}
}
TY - JOUR AU - Kiniger, Bernhard AU - Vexler, Boris TI - A priori error estimates for finite element discretizations of a shape optimization problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1733 EP - 1763 VL - 47 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013086/ DO - 10.1051/m2an/2013086 LA - en ID - M2AN_2013__47_6_1733_0 ER -
%0 Journal Article %A Kiniger, Bernhard %A Vexler, Boris %T A priori error estimates for finite element discretizations of a shape optimization problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1733-1763 %V 47 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013086/ %R 10.1051/m2an/2013086 %G en %F M2AN_2013__47_6_1733_0
Kiniger, Bernhard; Vexler, Boris. A priori error estimates for finite element discretizations of a shape optimization problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1733-1763. doi : 10.1051/m2an/2013086. http://www.numdam.org/articles/10.1051/m2an/2013086/
[1] Gascoigne: The finite element toolkit. http://www.gascoigne.uni-hd.de/
[2] Rodobo: A c++ library for optimization with stationary and nonstationary pdes. http://rodobo.uni-hd.de/
[3] and , Solvability of the Dirichlet problem for second-order elliptic equations in a convex domain. Differentsial′nye Uravneniya 28 (1992) 806-818, 917. | MR | Zbl
[4] and , A Primer of Nonlinear Analysis, vol. 34, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1993). | MR | Zbl
[5] , Finite Elemente, Springer-Verlag (2007). | Zbl
[6] and , Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybernet. 31 (2002) 695-712. | MR | Zbl
[7] and , A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53 (2012) 173-206. | MR | Zbl
[8] and , Controllability of an elliptic equation and its finite difference approximation by the shape of the domain. Numer. Math. 95 (2003) 63-99. | MR | Zbl
[9] and , Finite-element approximation of 2D elliptic optimal design. J. Math. Pures Appl. 85 (2006) 225-249. | MR | Zbl
[10] and , Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441-463. | MR | Zbl
[11] , , and , On convergence in elliptic shape optimization. SIAM J. Control Optim. 46 (2007) 61-83 (electronic). | MR
[12] , Elliptic problems in nonsmooth domains, vol. 24, Monographs and Studies in Mathematics, Pitman. Advanced Publishing Program, Boston, MA (1985). | MR | Zbl
[13] and , Introduction to shape optimization. Theory, approximation, and computation, vol. 7, Advances in Design and Control, Society for Industrial and Applied Mathematics SIAM. Philadelphia, PA (2003). | MR | Zbl
[14] and , Finite element approximation for optimal shape, material and topology design. John Wiley & Sons Ltd., Chichester, 2nd edition (1996). | MR | Zbl
[15] and , Lagrange multiplier approach to variational problems and applications, vol. 15, Advances in Design and Control, Society for Industrial and Applied Mathematics. SIAM, Philadelphia, PA (2008). | MR | Zbl
[16] and , The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. (N.S.) 4 (1981) 203-207. | MR | Zbl
[17] and , The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995) 161-219. | MR | Zbl
[18] , The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain. Czechoslovak Math. J. 14 (1964) 386-393. | MR | Zbl
[19] and , Numerical gradients for shape optimization based on embedding domain techniques. Comput. Optim. Appl. 18 (2001) 95-114. | MR | Zbl
[20] , A comparison of numerical methods for optimal shape design problems. Optim. Methods Softw. 10 (1999) 497-537. | MR | Zbl
[21] , Newton's method for a class of optimal shape design problems. SIAM J. Optim. 10 (2000) 503-533 (electronic). | Zbl
[22] , Sur la coercivité des formes sesquilinéaires, elliptiques. Rev. Roumaine Math. Pures Appl. 9 (1964) 47-69. | MR | Zbl
[23] and , Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38 (1982) 437-445. | MR | Zbl
[24] , Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998) 176-201. | MR | Zbl
[25] , Shape optimization for semi-linear elliptic equations based on an embedding domain method. Appl. Math. Optim. 49 (2004) 183-199. | MR | Zbl
[26] and , Introduction to shape optimization, Shape sensitivity analysis, vol. 16, Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1992). | Zbl
[27] , Optimale Steuerung partieller Differentialgleichungen, Vieweg+Teubner (2009).
Cité par Sources :
